Test: Combinatorics - Computer Science Engineering (CSE) MCQ

Concept:
Electric Generator:

• A generator is a machine that converts mechanical energy into electrical power for use in an external circuit. ​
• An electromechanical energy conversion device converts electrical energy into mechanical energy or mechanical to electrical energy
• An electromechanical energy conversion device converts electrical energy into mechanical energy or mechanical to electrical energy. The conversion can take place via any medium, electrical, or magnetic medium.
• Generally, the magnetic field is used as the coupling medium between electrical and mechanical mediums because the energy-storing capacity of the magnetic field is much higher than the electric field. It is a reversible process
• A prime mover is a machine that converts energy into work and examples of such machines are the gas turbine, steam turbine, reciprocating internal combustion engine, and hydraulic turbine

Given:
In an A.P. the m times of mth term is equal to n times the nth term

Concept used:
nth term of A.P = a_n = a + (n - 1)d

Calculation:
According to question,
⇒ m × a_m = n × a_n
⇒ m[a + (m - 1)d] = n[a + (n - 1)d] 
ma + m(m - 1)d = na + n(n - 1)d 
ma + m²d - md = na + n²d - nd
a(m - n) + (m² - n²)d = d(m - n) 
(m - n)[a + d(m + n - 1] = 0
∴ [a + d(m + n - 1] = 0 = a_{m + n}
∴ a_{m + n} = 0
∴ (m + n)^th term will be 0.

Given:


Concept:
Sum of an infinite G.P. series,  when |r| < 1
Sum of an A.P. series,   where, l = last term of series

Calculation:
For Series 1, a = 1, and r = 1/2

⇒ S₁ = 2
Similarly, for Series 2, a = 2 and r = 1/3

⇒ S₂ = 3
Similarly,
S₃ = 4
S₄ = 5, ...
S_n = n + 1
So, S₁, S₂, S₃, ... , Sn is an Arithmetic Progression (A.P.),
For A.P.,
a = S1 = 2
n = n
l = Sn = n + 1,
Therefore,

Given:


Analysis:


As the series converges to a positive number
∴ Positive roots of x² – x – 7 is the solution.

Given,
The given AP is 13, 11, 9……

Formula:
T_nt = a + (n – 1)d
a = first term
d = common term

Calculation:
a = 13
d = 11 – 13
d = (-2)
T₅₀ = 13 + (50 – 1) × (-2)
⇒ T₅₀ = 13 + 49 × (-2)
⇒ T₅₀ = 13 – 98
∴ T₅₀ = -85

Concept:
A p-series is a specific type of infinite series. It's a series of the form as shown below,

where p can be any real number greater than zero.
Notice that in this definition n will always take on positive integer values, and the series is an infinite series because it's a sum containing infinite terms.
There are infinitely many p-series because you have infinite choices for p. Each time you choose a different value for p you create another p-series.
With p-series,
If p > 1, the series will converge, or in other words, the series will add up to a specific numerical value.
If 0 < p ≤ 1, the series will diverge, which means that the series won't add up to a specific numerical value.

Option 1:
The series 
Proof: 
We know n² > n(n-1)

Expanding we get 

It follows that an is bounded from above and hence convergent.

Option 2 and 4:
A sequence {an} of real numbers is called a Cauchy sequence if for each ∈ > 0 there is a number N ∈ N so that if m, n > N then |an − am| < ∈.
If a real sequence {an} converges, then for every ε > 0, there exists N ∈ N such that |an − am| < ε ∀ n,m ≥ N
Convergent sequences are Cauchy sequences.
A Cauchy sequence of real numbers is bounded.
A sequence is a convergent sequence if and only if it is a Cauchy sequence.

Option 3:
A sequence of real numbers need not be bounded, a sequence can be bounded or not bounded.

Formula used:
Sum of A.P. = n/2{first term + last term}

Calculation:
Number of terms = n = 12
⇒ S_n = 12/2{5 + 38}
⇒ S_n = 6{43}
⇒ S_n = 258

Given:
A. M. of 6, 7, 9, x is 10, 

Concept used:
Mean = sum of all observation/total number of observations

Calculation:
A. M. of 6, 7, 9, x is 10, 
⇒ 10 = (6 + 7 + 9 + x)/4
⇒ 40 = 22 + x
∴ x = 18

Given series is,

By ratio test, the given series converges for |x| < 1 and diverges for |x| > 1
Let us examine the series for x = ± 1
For x = 1, the series reduces to

This is an alternating series and is convergent.
For x = -1 the series becomes

This is a divergent series as can be seen by comparison with P-series with P = 1
Hence the given series is converges for -1 < x < 1